Witch of Agnesi

Stage: 5 Challenge Level:

Why do this problem?
The problem gives practice in the usual techniques for cuve
sketching (considering symmetry, finding turning points, looking
for asymptotes). It also introduces the idea of a family of
curves.

Possible approach
Suggest different members of the class sketch the different graphs
(for $a=1$, $2$ and $3$). Have a class discussion about the results
they find.

Key question
Will the graphs have a similar shape for all values of $a$?

What about negative values of $a$?

Possible extension
If the class can differentiate simple functions defined
parametrically or implicitly then they could also try:
Squareness and
Folium of Descartes .

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NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.